Harmonic Functions in Slabs and Half-Spaces

Abstract

The usual solution to the Dirichlet problem for the Laplace equation Δu = 0 in the slab \({\mathbb R}^n \times (a,b)\), where −∞ < a < b < ∞, and the half-space \({\mathbb R}^n \times (0,\infty )\) involves convolution of the data with a Poisson kernel. Interestingly, the class of distributions which is convolvable with the natural Poisson kernel Q for the slab is considerably wider than that which is convolvable with the classical Poisson kernel P for the half-space. We investigate this curious phenomenon and observe that arbitrary tempered distributions can be convolved with Q, resulting in functions harmonic in the slab with no greater than polynomial growth in the interior and distributionally bounded on hyperplanes parallel to the boundary. Conversely, we show that all harmonic functions in the slab which enjoy no greater than polynomial growth in the interior and are distributionally bounded on hyperplanes parallel to the boundary can be characterized as Poisson integrals of tempered distributions. In the case of the half-space we observe that the classical Poisson kernel can be modified so that the result is applicable to all tempered distributions and gives rise to harmonic functions in the half-space with the prescribed boundary values. In both cases if the boundary data is given by polynomials then so is the resulting harmonic function. In the appendix we record some additional properties of the kernel Q and offer several pertinent comments and observations.